Q1.
Can these quantities ever be equal?
Answer
Yes. The total distance travelled and the magnitude of displacement are equal whenever the object moves in one direction along a straight line and never turns back.
In Fig. 4.4 the athlete does turn back, so the two differ:
between t = 0 s and t = 16 s:
total distance travelled = OA + AB = 100 m + 60 m = 160 m
magnitude of displacement = OB = 40 m
160 m ≠ 40 m
total distance travelled = OA + AB = 100 m + 60 m = 160 m
magnitude of displacement = OB = 40 m
160 m ≠ 40 m
But between t = 0 s and t = 10 s she runs only from O to A, without reversing:
total distance travelled = OA = 100 m
magnitude of displacement = OA = 100 m → equal
magnitude of displacement = OA = 100 m → equal
Why it happens: distance adds up every bit of path, treating forward and backward alike. Displacement subtracts backward motion from forward motion. As long as there is no backward motion to subtract, the two sums are identical. The moment the object reverses, the return path adds to the distance but cancels part of the displacement — so from then on the magnitude of displacement is always less than the distance travelled.